Problem 247 - A Complete Resource Book in Chemistry for JEE Main

Question

Find the \(\mathrm{pH}\) of a \(10^{-2} \mathrm{M}\) solution of sodium salt of substituted benzoic acid if the dissociation constant of substituted benzoic acid is \(1 \times 10^{-6}\) at \(298 \mathrm{~K}\)

Step-by-Step Solution

Answer

The \(\text{pH}\) of the solution is 9.

Step 1: Understand the Problem

We are given a sodium salt of a substituted benzoic acid with a concentration of \(10^{-2} \text{ M}\). The dissociation constant (\(K_a\)) of the substituted benzoic acid is \(1 \times 10^{-6}\). Our task is to find the \(\text{pH}\) of this solution.

Step 2: Relate the Dissociation Constants

For a salt derived from a weak acid and a strong base, the dissociation constant of the weak acid \(K_a\) is used to determine the base dissociation constant \(K_b\) of its conjugate base. This is done using the relation:\[ K_w = K_a \, K_b \]where \(K_w\) is the ionic product of water (\(1 \times 10^{-14}\) at 298 K).

Step 3: Calculate Kb (Base Dissociation Constant)

Using the relation between \(K_a\) and \(K_b\), calculate \(K_b\) as follows:\[ K_b = \frac{K_w}{K_a} = \frac{1 \times 10^{-14}}{1 \times 10^{-6}} = 1 \times 10^{-8}\].

Step 4: Set up the Equation for Hydroxide Ion Concentration

The concentration of hydroxyl ions \([OH^-]\) in the solution of the salt can be determined using the expression for \(K_b\):\[ K_b = \frac{[OH^-]^2}{C - [OH^-]} \]where \([OH^-]\) is the hydroxide ion concentration and \(C\) is the concentration of the salt (\(10^{-2}\) M). Assuming \([OH^-] \ll C\), simplify this to \([OH^-] \approx \sqrt{K_b \cdot C}\).

Step 5: Calculate the Hydroxide Ion Concentration

Using the approximation, find \([OH^-]\):\[ [OH^-] \approx \sqrt{1 \times 10^{-8} \cdot 10^{-2}} = \sqrt{1 \times 10^{-10}} = 1 \times 10^{-5} \text{ M}\].

Step 6: Determine the pOH

Calculate \(\text{pOH}\) using the hydroxide ion concentration:\[ \text{pOH} = -\log[OH^-] = -\log(1 \times 10^{-5}) = 5\].

Step 7: Convert pOH to pH

Finally, convert \(\text{pOH}\) to \(\text{pH}\) using the relation:\[ \text{pH} = 14 - \text{pOH} = 14 - 5 = 9 \].

Key Concepts

Weak Acid-Strong Base Salt

A weak acid-strong base salt, such as sodium benzoate, forms when a weak acid reacts with a strong base. This results in a salt that can dissolve in water and dissociate into its respective ions. For example, imagine sodium benzoate dissolving to produce benzoate ions which are the conjugate base of the weak benzoic acid.
The interaction in water is interesting due to the weak acid's partial dissociation. While the strong base, like sodium hydroxide, fully dissociates, the presence of the weak acid component influences the pH due to its limited tendency to release hydrogen ions completely.

Dissociation Constant

The dissociation constant (\( K_a \)) provides insight into the strength of an acid in aqueous solution. It measures the degree to which the acid separates into ions. For weak acids, \( K_a \) values are less than 1, reflecting their incomplete ionization in water.
To find \( K_b \) of the conjugate base in situations involving weak acid-strong base salts, we use the relationship \( K_w = K_a \times K_b \), where \( K_w \) is the ionic product of water. This allows us to determine the extent to which the conjugate base can accept protons.

Ionic Product of Water

The ionic product of water, \( K_w \), is a fundamental concept in chemistry that represents the self-ionization of water. At 25°C (298 K), \( K_w \) is typically \( 1 \times 10^{-14} \).
This constant is crucial for calculations as it links the acidic and basic dissociation constants (\( K_a \) and \( K_b \)). By understanding \( K_w \), we can account for the behavior of both hydrogen and hydroxide ions and predict the neutrality or alkalinity of solutions in equilibrium.

Hydroxide Ion Concentration

The hydroxide ion concentration, \([OH^-]\), is essential for determining the basicity of a solution. When a weak acid-strong base salt dissolves, hydroxide ions can form, shifting the solution's pH. The dissociation constant of the conjugate base allows us to estimate \([OH^-]\) using:

• \([OH^-] \approx \sqrt{K_b \cdot C}\)

This equation regards \( K_b \) and the initial concentration \( C \) of the salt, assuming the \([OH^-]\) contribution is small. Knowing \([OH^-]\) helps us compute pOH , aiding in the final conversion to pH .